The Schubert calculus, braid relations, and generalized cohomology

Let X be the flag variety of a compact Lie group and let h* be a complex-oriented generalized cohomology theory. We introduce operators on h*(X) which generalize operators introduced by Bernstein, Gel'fand, and Gel'fand for rational cohomology and by Demazure for K-theory. Using the Becker-Gottlieb transfer, we give a formula for these operators, which enables us to prove that they satisfy braid relations only for the two classical cases, thereby giving a topological interpretation of a theorem proved by the authors and extended by Gutkin. One of the central issues in Lie theory is the geometry of the flag variety associated to a compact Lie group G. An important problem concerning the flag variety is the Schubert calculus, which studies the ring structure of the cohomology of the flag variety. Work initiated by Borel, Bott and Kostant, which culminated in a paper by Bernstein, Gel'fand and Gel'fand [BGG], gave a complete solution to the problem. Demazure studied the same problem for K-theory. Moreover, these techniques have been generalized to the Kac-Moody situation by Kac-Peterson, Kostant-Kumar, and others. This work has focussed on algebro-geometric properties of the flag variety. Here, on the other hand we study the flag variety from the point of view of algebraic topology. As a consequence, not only do we recover the classical results described above, but we extend these results to a certain class of cohomology theories-those which are complex-oriented. Examples of complex-oriented theories include ordinary cohomology, K-theory, complex cobordism, and elliptic cohomology. Since the context we have chosen in very general, the proofs are universal and are often simpler than the classical arguments. In the work of BGG, a crucial role is played by operators Ai associated to each simple reflection si of the Weyl group of G (defined by Demazure in K-theory). These operators Ai satisfy the braid relations, which are the relations between pairs of simple reflections. In this paper, we generalize the A, to give operators D, acting on h*(G/T) for any complex-oriented theory h*. We prove that braid relations are satisfied only for cohomology theories with the formal group law of rational cohomology or of K-theory (Theorem Received by the editors June 21, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 55N20, 57T15. The second author was supported by an NSF graduate fellowship. (3 1990 American Mathematical Society 0002-9947/90 $1.00 + $.25 perpage